**3. Optimal Homotopic Solutions**

The initial guesses and linear operators for the construction of series solutions are

$$f(\eta) = A\eta + (1 - A)(1 - \operatorname{Exp}(-\eta), \,\theta\_0(\eta) = (1 - S)\operatorname{Exp}(-\eta), \,\phi\_0(\eta) = (1 - St)\operatorname{Exp}(-\eta), \tag{20}$$

$$\mathcal{L}\_f\left(f\right) = \frac{d^3f}{d\eta^3} - \frac{df}{d\eta}, \quad \mathcal{L}\_\theta\left(\theta\right) = \frac{d^2\theta}{d\eta^2} - \theta, \quad \mathcal{L}\_\phi\left(\theta\right) = \frac{d^2\theta}{d\eta^2} - \theta,\tag{21}$$

with

$$\mathcal{L}\_f \left[ A\_1 + A\_2 \exp(\eta) + A\_3 \exp(-\eta) \right] = 0,\tag{22}$$

$$\mathcal{L}\_{\theta} \left[ A\_4 \exp(\eta) + A\_5 \exp(-\eta) \right] = 0,\tag{23}$$

$$\mathcal{L}\_{\Phi} \left[ A\_{\theta} \exp(\eta) + A\_{7} \exp(-\eta) \right] = 0,\tag{24}$$

where *Ai* (*i* = 1, 2, . . . , 7) are the arbitrary constants.
